On some algebraic properties of CM-types of CM-fields and their reflexes

TL;DR

This paper explores new algebraic properties of CM-types and their reflex fields, revealing a combinatorial structure that leads to three theorems connecting abelian extensions, character identities, and quadratic forms in complex multiplication theory.

Contribution

It introduces a novel combinatorial structure on reflex fields of CM-fields and proves three theorems relating abelian extensions, character identities, and quadratic forms.

Findings

Reflex fields have a specific combinatorial structure.

Generalization of Wei's result on abelian extensions from CM-type abelian varieties.

Establishment of a character identity for Artin L-functions of CM-fields.

Abstract

The purpose of this paper is to show that the reflex fields of a given CM-field is equipped with a certain combinatorial structure that has not been exploited yet. We prove three theorems using this structure; the first theorem is on the abelian extension generated by the moduli and the b-torsion points of abelian varieties of CM-type, for any natural number b. It is a generalization of the result by Wei on the abelian extension obtained by the moduli and all the torsion points. The second theorem gives a character identity of the Artin L-function of a CM-field K and the reflex fields of K. The character identity pointed out by Shimura follows from this. The third theorem states that some Pfister form is isomorphic to the orthogonal sum of Tr(\bar{a}a) (\bar{a} is the complex conjugation of a) defined on a direct sum of reflex fields. This result suggests that the theory of complex multiplication on abelian varieties has a relationship with the multiplicative forms in higher dimension.